Encyclopedia of Microtonal Music Theory

chromatic genus

[Joe Monzo]

In ancient Greek theory, one of the three basic types of genus. It had a characteristic interval of approximately a minor 3rd at the top of the tetrachord, then 2 successive intervals of approximately a semitone at the bottom, making up the 4/3 perfect 4th.

Below is a graph showing the comparative structures of tetrachords for the chromatic genus as explained by various ancient theorists:

As with the classic Pythagorean diatonic described by Philolaus, Thrasyllus's chromatic, if extended beyond the single tetrachord, could be tuned using the method of "tuning by concords", using 3:2 perfect-5ths and 4:3 perfect-4ths up and down from the starting-note mese.

Archytas measured the upper boundary of his pyknon as a limma below that of the diatonic, making his lichanos the same Pythagorean 32:27 trihemitone below the top note (mese) of the tetrachord as that described by Thrasyllus. This lichanos is also a 9:8 "tone" above the bottom note hypate; Archytas's chromatic pyknon thus has the string-length proportion 8:9; multiplied by 28 and with the middle note inserted the proportions are 224:243:252, with the respective arithmetic differences of 19 and 9.

Archytas's parhypate is the same as in his diatonic genus and enharmonic genus. (For a discussion of Archytas's idiosyncratic use of 28:27 as the lowest interval in all three of his genera, see the explanation of his diatonic genus.)

For his characteristic interval at the top of the tetrachord, Eratosthenes replaced the Pythagorean "trihemitone" with the 5-limit 6:5 JI minor-3rd. The lichanos is therefore also a 5-limit 10:9 "lesser tone" above the bottom note hypate. Eratosthenes then multiplied these terms by 2 and divided this pyknon by its arithmetic mean to get the superparticular proportions 18:19:20. Note that his parhypate is the "enneadecimal major-3rd" which is very similar to the Pythagorean 81:64 "ditone", making the bottom part of the tetrachord resemble that of Thrasyllus.

Didymus retained the 5-limit 6:5 minor-3rd "characteristic interval" introduced by Eratosthenes, then altered the other two ratios so that his entire chromatic genus is 5-limit, introducing the 25:24 "chromatic semitone" in the middle and (as he had done in his diatonic genus) the 16:15 "diatonic semitone" at the bottom of the tetrachord.

In his "tense" version of the chromatic genus, Ptolemy used the narrow 7:6 "septimal minor-3rd" as his "characteristic interval" at the top of the tetrachord, thus simultaneously placing his lichanos a wide 8:7 "septimal tone" above the bottom note hypate. The middle interval is the 12:11 "undecimal neutral 2nd", functioning here as a very wide semitone.

For his "relaxed" version of the chromatic genus, Ptolemy chose for his "characteristic interval" the same 5-limit 6:5 "minor-3rd" as Eratosthenes and Didymus. Note Ptolemy's placement of parhypate a wide 9:7 "septimal major-3rd" below the top note mese, as in Archytas's version of the chromatic genus, and thus giving the same 28:27 as Archytas for his lowest interval. The middle interval is therefore the 15:14 "septimal semitone".